{"id":90363,"date":"2025-06-01T10:27:05","date_gmt":"2025-06-01T10:27:05","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=90363"},"modified":"2025-06-01T10:27:05","modified_gmt":"2025-06-01T10:27:05","slug":"if-a-circle-and-a-square-have-the-same-perimeter-then","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/if-a-circle-and-a-square-have-the-same-perimeter-then\/","title":{"rendered":"If a circle and a square have the same perimeter, then"},"content":{"rendered":"<p>If a circle and a square have the same perimeter, then<\/p>\n<p>[amp_mcq option1=&#8221;their areas are equal&#8221; option2=&#8221;the area of the circle is greater than the area of the square&#8221; option3=&#8221;the area of the square is greater than the area of circle&#8221; option4=&#8221;the area of the circle is two times the area of the square&#8221; correct=&#8221;option2&#8243;]<\/p>\n<div class=\"psc-box-pyq-exam-year-detail\">\n<div class=\"pyq-exam\">\n<div class=\"psc-heading\">This question was previously asked in<\/div>\n<div class=\"psc-title line-ellipsis\">UPSC CAPF &#8211; 2019<\/div>\n<\/div>\n<div class=\"pyq-exam-psc-buttons\"><a href=\"\/pyq\/pyq-upsc-capf-2019.pdf\" target=\"_blank\" class=\"psc-pdf-button\" rel=\"noopener\">Download PDF<\/a><a href=\"\/pyq-upsc-capf-2019\" target=\"_blank\" class=\"psc-attempt-button\" rel=\"noopener\">Attempt Online<\/a><\/div>\n<\/div>\n<section id=\"pyq-correct-answer\">\nIf a circle and a square have the same perimeter, the area of the circle is greater than the area of the square.<br \/>\n<\/section>\n<section id=\"pyq-key-points\">\nLet the perimeter of both the circle and the square be $P$.<br \/>\nFor a square with side length $s$, the perimeter is $4s=P$, so $s = P\/4$. The area of the square is $A_{\\text{square}} = s^2 = (P\/4)^2 = P^2\/16$.<br \/>\nFor a circle with radius $r$, the perimeter is $2\\pi r=P$, so $r = P\/(2\\pi)$. The area of the circle is $A_{\\text{circle}} = \\pi r^2 = \\pi (P\/(2\\pi))^2 = \\pi (P^2\/(4\\pi^2)) = P^2\/(4\\pi)$.<br \/>\n<\/section>\n<section id=\"pyq-additional-information\">\nTo compare the areas, we compare $P^2\/16$ and $P^2\/(4\\pi)$. This is equivalent to comparing $1\/16$ and $1\/(4\\pi)$.<br \/>\nSince $\\pi \\approx 3.14159$, $4\\pi \\approx 12.566$.<br \/>\nComparing $1\/16$ and $1\/12.566$. Since $16 > 12.566$, it follows that $1\/16 < 1\/12.566$.\nTherefore, $A_{\\text{square}} < A_{\\text{circle}}$. The area of the circle is greater than the area of the square. This is a general geometric principle: among all planar shapes with the same perimeter, the circle has the largest area.\n<\/section>\n","protected":false},"excerpt":{"rendered":"<p>If a circle and a square have the same perimeter, then [amp_mcq option1=&#8221;their areas are equal&#8221; option2=&#8221;the area of the circle is greater than the area of the square&#8221; option3=&#8221;the area of the square is greater than the area of circle&#8221; option4=&#8221;the area of the circle is two times the area of the square&#8221; correct=&#8221;option2&#8243;] &#8230; <\/p>\n<p class=\"read-more-container\"><a title=\"If a circle and a square have the same perimeter, then\" class=\"read-more button\" href=\"https:\/\/exam.pscnotes.com\/mcq\/if-a-circle-and-a-square-have-the-same-perimeter-then\/#more-90363\">Detailed Solution<span class=\"screen-reader-text\">If a circle and a square have the same perimeter, then<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1085],"tags":[1119,1102],"class_list":["post-90363","post","type-post","status-publish","format-standard","hentry","category-upsc-capf","tag-1119","tag-quantitative-aptitude-and-reasoning","no-featured-image-padding"],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.2 (Yoast SEO v23.3) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>If a circle and a square have the same perimeter, then<\/title>\n<meta name=\"description\" content=\"If a circle and a square have the same perimeter, the area of the circle is greater than the area of the square. Let the perimeter of both the circle and the square be $P$. For a square with side length $s$, the perimeter is $4s=P$, so $s = P\/4$. The area of the square is $A_{text{square}} = s^2 = (P\/4)^2 = P^2\/16$. For a circle with radius $r$, the perimeter is $2pi r=P$, so $r = P\/(2pi)$. The area of the circle is $A_{text{circle}} = pi r^2 = pi (P\/(2pi))^2 = pi (P^2\/(4pi^2)) = P^2\/(4pi)$.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/if-a-circle-and-a-square-have-the-same-perimeter-then\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"If a circle and a square have the same perimeter, then\" \/>\n<meta property=\"og:description\" content=\"If a circle and a square have the same perimeter, the area of the circle is greater than the area of the square. Let the perimeter of both the circle and the square be $P$. For a square with side length $s$, the perimeter is $4s=P$, so $s = P\/4$. The area of the square is $A_{text{square}} = s^2 = (P\/4)^2 = P^2\/16$. For a circle with radius $r$, the perimeter is $2pi r=P$, so $r = P\/(2pi)$. The area of the circle is $A_{text{circle}} = pi r^2 = pi (P\/(2pi))^2 = pi (P^2\/(4pi^2)) = P^2\/(4pi)$.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/if-a-circle-and-a-square-have-the-same-perimeter-then\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T10:27:05+00:00\" \/>\n<meta name=\"author\" content=\"rawan239\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rawan239\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"If a circle and a square have the same perimeter, then","description":"If a circle and a square have the same perimeter, the area of the circle is greater than the area of the square. Let the perimeter of both the circle and the square be $P$. For a square with side length $s$, the perimeter is $4s=P$, so $s = P\/4$. The area of the square is $A_{text{square}} = s^2 = (P\/4)^2 = P^2\/16$. For a circle with radius $r$, the perimeter is $2pi r=P$, so $r = P\/(2pi)$. The area of the circle is $A_{text{circle}} = pi r^2 = pi (P\/(2pi))^2 = pi (P^2\/(4pi^2)) = P^2\/(4pi)$.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/exam.pscnotes.com\/mcq\/if-a-circle-and-a-square-have-the-same-perimeter-then\/","og_locale":"en_US","og_type":"article","og_title":"If a circle and a square have the same perimeter, then","og_description":"If a circle and a square have the same perimeter, the area of the circle is greater than the area of the square. Let the perimeter of both the circle and the square be $P$. For a square with side length $s$, the perimeter is $4s=P$, so $s = P\/4$. The area of the square is $A_{text{square}} = s^2 = (P\/4)^2 = P^2\/16$. For a circle with radius $r$, the perimeter is $2pi r=P$, so $r = P\/(2pi)$. The area of the circle is $A_{text{circle}} = pi r^2 = pi (P\/(2pi))^2 = pi (P^2\/(4pi^2)) = P^2\/(4pi)$.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/if-a-circle-and-a-square-have-the-same-perimeter-then\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T10:27:05+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/if-a-circle-and-a-square-have-the-same-perimeter-then\/","url":"https:\/\/exam.pscnotes.com\/mcq\/if-a-circle-and-a-square-have-the-same-perimeter-then\/","name":"If a circle and a square have the same perimeter, then","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T10:27:05+00:00","dateModified":"2025-06-01T10:27:05+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"If a circle and a square have the same perimeter, the area of the circle is greater than the area of the square. Let the perimeter of both the circle and the square be $P$. For a square with side length $s$, the perimeter is $4s=P$, so $s = P\/4$. The area of the square is $A_{\\text{square}} = s^2 = (P\/4)^2 = P^2\/16$. For a circle with radius $r$, the perimeter is $2\\pi r=P$, so $r = P\/(2\\pi)$. The area of the circle is $A_{\\text{circle}} = \\pi r^2 = \\pi (P\/(2\\pi))^2 = \\pi (P^2\/(4\\pi^2)) = P^2\/(4\\pi)$.","breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/if-a-circle-and-a-square-have-the-same-perimeter-then\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/if-a-circle-and-a-square-have-the-same-perimeter-then\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/if-a-circle-and-a-square-have-the-same-perimeter-then\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"UPSC CAPF","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/upsc-capf\/"},{"@type":"ListItem","position":3,"name":"If a circle and a square have the same perimeter, then"}]},{"@type":"WebSite","@id":"https:\/\/exam.pscnotes.com\/mcq\/#website","url":"https:\/\/exam.pscnotes.com\/mcq\/","name":"MCQ and Quiz for Exams","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/exam.pscnotes.com\/mcq\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209","name":"rawan239","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","caption":"rawan239"},"sameAs":["https:\/\/exam.pscnotes.com"],"url":"https:\/\/exam.pscnotes.com\/mcq\/author\/rawan239\/"}]}},"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/90363","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/comments?post=90363"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/90363\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=90363"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=90363"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=90363"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}